B3.2 Calculate amount and principal with the compound-interest formula

Ontario Grade 11 Mathematics

Exponential Functions

Calculate the final amount and find the starting principal

Interest is money earned on savings or charged on a loan. With compound interest, interest is added to the balance at regular times. Later interest is calculated on the new balance, which includes earlier interest. This lesson focuses on using one formula to find either the final amount or the starting principal. A principal is the original amount invested or borrowed. An amount is the balance after interest has been added.

What you will learn

Identify the principal, interest rate, time, and number of compounding periods in a problem.
Use the compound-interest formula to calculate an amount.
Rearrange the formula to calculate the principal when the amount is known.
Check that calculator entries and final answers make sense.

1. Prerequisite bridge: percent and powers

A percent is a rate out of one hundred. To use a percent rate in a calculation, write it as a decimal. For example, an annual rate of 4% is

0.04

. The decimal form is useful because it can be added to

1

to make a growth factor:

1+0.04=1.04

.

A power is a short way to show repeated multiplication. For example,

1.04^3

means

1.04\times1.04\times1.04

. In compound interest, the power counts how many times the balance grows by the interest factor.

The time and the interest rate must use matching time units. If a rate is given per year, count the number of years for annual compounding. If interest is compounded several times per year, count all those compounding periods instead.

1+r

Convert a percent rate to a decimal before using it.
The exponent counts compounding periods, not necessarily years.
Match the rate period and the time count.

2. What the formula represents

The compound-interest formula gives the balance after interest has been added repeatedly. The amount depends on the starting principal, the interest rate per compounding period, and the total number of periods.

In the formula,

A

means the final amount, and

P

means the principal. The symbol

r

is the interest rate written as a decimal for each compounding period. The symbol

n

is the total number of compounding periods. The growth factor is

1+r

. Raising it to the power

n

repeats that growth for all periods.

For example, if an account earns 4% per year and interest is added once each year for three years, then the rate per period is

0.04

and there are three periods. If interest is added four times each year for three years, there are twelve periods. When the stated rate is annual and interest is compounded several times a year, divide the annual rate by the number of compounding periods per year to get the rate for one period.

To find the principal instead, divide the final amount by the growth factor raised to the number of periods. This reverses the repeated growth. Use this form when the final amount, rate, and number of periods are known.

A=P(1+r)^n

Use the rate for one compounding period in the formula.
The exponent is the total number of compounding periods.
The principal can be found by undoing the repeated growth with division.

3. Set up the values before calculating

Start by writing down what the problem gives and what it asks you to find. Decide how often interest is compounded. If compounding happens

m

times per year for

t

years, the total number of periods is

mt

. For an annual rate

i

, the rate per period is

i/m

. Both the division and the multiplication matter: one converts the rate, and the other counts the periods.

A table can help keep the information organized. For the principal calculation, use the same rate and period count as for the amount calculation. Do not change the compounding schedule when rearranging the formula.

Enter the full growth factor in parentheses before applying the power. Keep extra digits during the calculation, then round the final money amount to the nearest cent. Rounding too early can slightly change the result.

r=

\frac{i}{m}

, n=mt

Find the rate per period and total periods separately.
Use parentheses around the growth factor on a calculator.
Round money only at the end.

4. Guided example and application

Suppose CAD 1,200 is invested at an annual rate of 4%, compounded quarterly, for three years. First find the rate per quarter: divide the annual decimal rate by four. Then find the number of quarters by multiplying four by three. Use these values to calculate the amount.

To see how the reverse calculation works, imagine the known amount is the result from this investment, and the principal is missing. Divide that amount by the same growth factor and power. The result should return the starting principal, apart from any small difference caused by rounding.

This reverse check is useful when a problem asks how much was originally invested to reach a stated balance. It does not require a new interest rule; it uses the same compound-interest relationship with the principal isolated.

P=\frac{A}{(1+r)^n}

Quarterly compounding means four periods per year.
Use the same growth factor and period count for amount and principal calculations.
A reverse calculation can check a result.

Information for the quarterly investment

Quantity	Value	Meaning
Principal	CAD 1,200	Starting investment
Annual rate	4%	Rate before adjusting for quarterly periods
Rate per period	$0.01$	One quarter's rate
Number of periods	12	Four periods per year for three years

Worked example

Amount first, then principal

An investment of CAD 1,200 earns an annual interest rate of 4%, compounded quarterly, for three years. Find the final amount. Then use that final amount to find the principal, as if the principal were unknown.

Convert the rate
The annual rate is $0.04$ as a decimal. Since interest is compounded four times per year, divide by four to get the rate for one quarter.
$r=\frac{0.04}{4}=0.01$
Count the periods
There are four quarters in each year and the investment lasts three years. Multiply to find the total number of compounding periods.
$n=4(3)=12$
Calculate the amount
Substitute the principal, rate per quarter, and total periods into the compound-interest formula. The growth factor is $1.01$ . Keep the full calculator value until rounding the final amount to cents.
$A=1200(1.01)^{12}\approx 1352.19$
Find the principal from the amount
Now treat the rounded final amount as known. Divide by the same growth factor raised to the same number of periods. This reverses the growth and gives approximately the original investment.
$P=\frac{1352.19}{(1.01)^{12}}\approx 1200$

Answer: The final amount is approximately CAD 1,352.19. The principal is approximately CAD 1,200.00.

Check: The amount is greater than the principal because a positive interest rate increases the balance. Using the unrounded amount in the reverse calculation returns exactly CAD 1,200.

Common mistakes and how to avoid them

Using

4

as the rate instead of converting 4% to a decimal.

Correction: Write the percent as a decimal first. For example,

4\%=0.04

.

Using the number of years as the exponent when interest is compounded more than once per year.

Correction: Count every compounding period. For quarterly compounding over three years, there are twelve periods.

Using the annual rate as the rate for each quarter.

Correction: Divide the annual decimal rate by four for quarterly compounding.

Multiplying the amount by the growth factor to find the principal.

Correction: Divide the amount by the growth factor raised to the total number of periods.

Rounding the growth factor or intermediate values too early.

Correction: Keep the calculator value during the calculation and round the final money amount to cents.

Lesson summary

Convert the annual percent rate to a decimal.
Adjust the rate to match the compounding period and count all periods.
Use the compound-interest formula to find the amount.
Divide the amount by the repeated growth factor to find the principal.
Round the final money value to the nearest cent.

Check your understanding

Question 1

CAD 500 is invested at 5% per year, compounded annually, for two years. What is the amount?

CAD 525.00
CAD 551.25
CAD 550.00
CAD 552.50

Show answer and explanation

CAD 551.25

There are two periods and the rate per period is

0.05

. The calculation is

500(1.05)^2=551.25

, so the amount is CAD 551.25.

Question 2

An amount of CAD 1,210 is reached after two years at 10% per year, compounded annually. What was the principal?

CAD 1,000
CAD 1,100
CAD 1,210
CAD 1,331

Show answer and explanation

CAD 1,000

There are two periods. Divide by the growth factor for both periods:

1210/(1.10)^2=1000

. The principal was CAD 1,000.

Question 3

An annual rate is 8% and interest is compounded monthly. What rate should be used for one compounding period?

$0.08$
$0.08/12$
$0.08(12)$
$8/12$

Show answer and explanation

0.08/12

Convert the annual rate to decimal form, then divide by twelve months. The monthly rate is

0.08/12

.

Key terms

Amount: The balance after compound interest has been added.
Compound interest: Interest calculated on a balance that includes previously added interest.
Compounding period: One time interval after which interest is added to the balance.
Principal: The original amount invested or borrowed.
Rate per period: The interest rate written as a decimal for one compounding period.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B3.2. It is a study resource, not an official curriculum publication.

Official curriculum reference

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B3.2 · Calculate amount and principal with the compound-interest formula

Ontario Grade 11 Mathematics

Calculate the final amount and find the starting principal

What you will learn

1. Prerequisite bridge: percent and powers

2. What the formula represents

3. Set up the values before calculating

4. Guided example and application

Information for the quarterly investment

Amount first, then principal

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson